The Earth's atmosphere acts as an electrical conductor due to the presence of free charge carriers generated by cosmic ray ionization. Assume that the free charge density remains constant across space and time and does not vary horizontally. The near-surface electric field is constant in time, strictly vertical, without any horizontal variations, and has a magnitude of 100 volts/meter.
By setting up appropriate equations and boundary conditions, determine the atmospheric electric field as a function of altitude. You may assume that the Earth's surface is perfectly flat for simplicity. Estimate the altitude dependence of atmospheric conductivity based on these conditions.
1
\( \sigma(z) \propto e^{-\frac{kT}{mgz}} \)
2
\( \sigma(z) \propto e^{-\frac{mgz}{kT}} \)
3
\( \sigma(z) \propto e^{\frac{mgz}{kT}} \)
4
\( \sigma(z) \propto e^{\frac{kT}{mgz}} \)